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Thread: MATH2111 Higher Several Variable Calculus

  1. #51
    Ancient Orator leehuan's Avatar
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    Re: Multivariable Calculus

    I'm so shit at this topic.






    ____________



    Edit: Why is LaTeX broken on this site again?

    Ok basically the main question is

    They had d/dv g(v+2t) = g'(v+2t)

    So why was it safe to integrate with respect to t to get 1/2 g(v+2t) again? I'm missing something elementary
    Last edited by leehuan; 14 Jun 2016 at 10:48 PM.

  2. #52
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    Re: Multivariable Calculus

    Quote Originally Posted by leehuan View Post
    I'm so shit at this topic.






    ____________



    Edit: Why is LaTeX broken on this site again?

    Ok basically the main question is

    They had d/dv g(v+2t) = g'(v+2t)

    So why was it safe to integrate with respect to t to get 1/2 g(v+2t) again? I'm missing something elementary
    They're integrating the derivative of a function, so they used FTC. (The 1/2 coming out due to reverse chain rule basically.)

    Note that (∂/∂v)(g(v+2t)) = g'(v+2t), where g' just means the derivative of the function g.

    E.g. if g(y) was cos(y), we'd have ∂/∂v (g(v+2t)) = ∂/∂v (cos(v+2t)) = -sin(v+2t) (i.e. g' evaluated at v+2t, which is what g'(v+2t) means).

    Then integrating -sin(v+2t), we'd get back to (1/2)cos(v+2t).
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  3. #53
    Ancient Orator leehuan's Avatar
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    Re: Multivariable Calculus

    Quote Originally Posted by InteGrand View Post
    They're integrating the derivative of a function, so they used FTC. (The 1/2 coming out due to reverse chain rule basically.)

    Note that (∂/∂v)(g(v+2t)) = g'(v+2t), where g' just means the derivative of the function g.

    E.g. if g(y) was cos(y), we'd have ∂/∂v (g(v+2t)) = ∂/∂v (cos(v+2t)) = -sin(v+2t) (i.e. g' evaluated at v+2t, which is what g'(v+2t) means).

    Then integrating -sin(v+2t), we'd get back to (1/2)cos(v+2t).
    Oops thanks.

    It makes sense with an example; I think I just got scared of too many variables here

  4. #54
    Supreme Member seanieg89's Avatar
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    Re: Multivariable Calculus

    Quote Originally Posted by leehuan View Post
    I'm so shit at this topic.






    ____________

    In the integrand we are applying g (a function of a single variable) to the expression v+2t.

    Partially differentiating g(v+2t) with respect to v and partially differentiating with respect to t just differ by a factor of 2, as computed using the chain rule. (Write h(v,t)=v+2t as a function from R^2 to R and carefully apply the chain rule to the composition (g o h) if you are still confused after this post.)

    So to summarise, Leibniz lets us differentiate the integral w.r.t. v by differentiating the integrand w.r.t. v. The new integrand can be replaced by the t-partial derivative of g(v+2t) up to the constant factor of 2 which we account for by division. Then we are just integrating the t-derivative of something w.r.t t which the FTC takes care of.
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    Supreme Member seanieg89's Avatar
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    Re: Multivariable Calculus

    Another exercise without much required knowledge.

    Suppose we have a smooth function .

    Suppose further, that given any starting point , we can find a smooth curve with and .

    (Intuitively, the vector field X is the infinitesimal generator of the curve c.)

    Under some simple assumptions, the function that maps a pair to where c is the curve starting at x, is a smooth function with smooth inverse for any fixed t.

    So we can consider how sets in R^n "flow". Of course, in general this will distort volume and other geometric quantities.

    One way of quantifying volume distortion locally about a point p is as follows:



    where |S| denotes the volume of a subset S in R^n, or equivalently, the integral of the constant function 1 over this set, which we assume is a well defined Riemann integral for the sets involved in this question and B(p,r) denotes the ball about p of radius r.

    (So E_X measures the limiting rate of change of volumes of small balls about p relative to the size of these balls.)

    1. Compute an expression for E_X(p) in terms of the components of X and it's derivatives.

    Followups to come if anyone answers this question.
    Last edited by seanieg89; 25 Jun 2016 at 12:48 PM.

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    Ancient Orator leehuan's Avatar
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    Re: Multivariable Calculus

    Quote Originally Posted by leehuan View Post
    So like, this question felt never ending and I aborted.

    Part b) is just the chain rule:



    Is there any shortcut to save me from computing several product and quotient rules in this one

    Aha thanks InteGrand, I gave this question much more thought today and finally got the proof out (one and a half months late :P )

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    Ancient Orator leehuan's Avatar
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    Re: Multivariable Calculus

    Three months later and I still don't know how to use total differential approximation. Correct answer = 0.21% (possibly approximated)




  8. #58
    -insert title here- Paradoxica's Avatar
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    Re: Multivariable Calculus

    Quote Originally Posted by leehuan View Post
    Three months later and I still don't know how to use total differential approximation. Correct answer = 0.21% (possibly approximated)



    Just find the equation of the tangent hyper-plane at the point ½absinC and plug in the given values...

    Here, the tangent hyperplane is given by:

    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Ancient Orator leehuan's Avatar
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    Re: Multivariable Calculus

    Quote Originally Posted by Paradoxica View Post
    Just find the equation of the tangent hyper-plane at the point ½absinC and plug in the given values...

    Here, the tangent hyperplane is given by:

    I'm asking for a friend. Can we just break it back down to appropriate use of the formula thanks.

  10. #60
    -insert title here- Paradoxica's Avatar
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    Re: Multivariable Calculus

    Quote Originally Posted by leehuan View Post
    I'm asking for a friend. Can we just break it back down to appropriate use of the formula thanks.
    it's the higher dimensional analogue of construction of a tangent at a point. fairly intuitive to understand if you ask me, and also I have no knowledge of what constitutes "appropriate use"
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  11. #61
    Ancient Orator leehuan's Avatar
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    Re: Multivariable Calculus

    I don't want my friends to just see the word "hyperplane" and run off. (Nor do I have enough of a grasp to translate it myself, even if it seems to make perfect sense to me)



    (think I meant basic. No idea where I got 'appropriate' from.)

  12. #62
    Rambling Spirit
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    Re: Multivariable Calculus



    Last edited by InteGrand; 21 Aug 2016 at 11:52 PM.
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    Ancient Orator leehuan's Avatar
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    Several Variable Calculus Marathon & Questions

    For some reason I lost the point of intersection (2,0):



    I equated the relevant r1 and r2 components but only got t=0

  14. #64
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    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post
    For some reason I lost the point of intersection (2,0):



    I equated the relevant r1 and r2 components but only got t=0
    Solve the system of equations

    t^2 - t = s + s^2 (1)
    t^2 + t = s - s^2 (2).
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    Ancient Orator leehuan's Avatar
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    Re: Several Variable Calculus

    Quote Originally Posted by InteGrand View Post
    Solve the system of equations

    t^2 - t = s + s^2 (1)
    t^2 + t = s - s^2 (2).
    Oh. I'll work on that right now but why was it necessary to introduce a new variable?

  16. #66
    Supreme Member seanieg89's Avatar
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    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post
    Oh. I'll work on that right now but why was it necessary to introduce a new variable?
    You have two parametric curves. Their points of intersection don't necessarily have the same parameter as a point on the first curve as they do as a point on the second curve.
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    Supreme Member seanieg89's Avatar
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    Re: Several Variable Calculus

    Eg, consider the lines (x,y)=(t,0), (x,y)=(2t,0).

    These lines coincide exactly, so every point on the x-axis is a point of intersection.

    Yet the only place where (t,0)=(2t,0) is at the origin.

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    Ancient Orator leehuan's Avatar
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    Re: Several Variable Calculus

    Excellent. Makes sense.
    ________________

    Find the angle between the two curves at the points of intersection.

    I'm having a dumb moment now. Which vectors are we taking the dot product of?

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    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post
    Excellent. Makes sense.
    ________________

    Find the angle between the two curves at the points of intersection.

    I'm having a dumb moment now. Which vectors are we taking the dot product of?
    Find the s and t values at the points of intersection and plug them into the derivatives of the parametric curves. This will give us the "direction vectors" of the curves at the points of intersection. Find the angle between these vectors.
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    Ancient Orator leehuan's Avatar
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    Re: Several Variable Calculus

    Strange... That's what I did so maybe there's an error in my computation.









    But the answer was arccos(0.8)

    EDIT: Ouch. I know what I did now.
    Last edited by leehuan; 28 Feb 2017 at 12:31 PM.

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    Re: Several Variable Calculus

    Last edited by InteGrand; 28 Feb 2017 at 12:32 PM. Reason: Typo
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    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post
    Strange... That's what I did so maybe there's an error in my computation.









    But the answer was arccos(0.8)
    You appear to have miscalculated the velocity vectors when subbing in the values of t and/or s (check the first components, noting you're subbing in s = 1 (not 2) and t = -1 (not -2)).
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  23. #73
    Ancient Orator leehuan's Avatar
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    Re: Several Variable Calculus







    Can be assumed: F=ma

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    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post






    Can be assumed: F=ma
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    Re: Several Variable Calculus

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